orthogonal projection onto a subspace

$\operatorname{proj}_W(\mathbf v)$ — summing single-vector projections onto an orthogonal basis of $W$ .

Given a subspace $W$ with orthogonal basis $\{\mathbf w_1,\dots,\mathbf w_k\}$ , the orthogonal projection of $\mathbf v$ onto $W$ is $\operatorname{proj}_W(\mathbf v)=\sum_i \frac{\mathbf v\cdot\mathbf w_i}{\mathbf w_i\cdot\mathbf w_i}\mathbf w_i$ — each term is last chapter's line-projection formula, computable independently because orthogonality lets each coefficient be read off from a single dot product. The result depends only on $W$ and $\mathbf v$ , not on which orthogonal basis is used.